Free binomial distribution calculator. Enter n, p, and k for the exact P(X=k), P(X<=k), P(X>=k), mean and variance, plus a bar chart of the distribution.
The binomial distribution models the number of successes in a fixed number of independent yes/no trials, where each trial succeeds with the same probability. Enter the number of trials n, the success probability p, and a number of successes k, and this calculator returns the exact probability of getting exactly k, at most k, and at least k successes, plus the distribution's mean and variance and a bar chart of the whole distribution.
A scenario is binomial when it meets four conditions, sometimes called Bernoulli trials: (1) there is a fixed number of trials n; (2) each trial has only two outcomes, "success" or "failure"; (3) the probability of success p is the same on every trial; and (4) the trials are independent of one another. Flipping a coin 20 times, counting defective items in a batch of 100, or counting how many of 12 free throws go in are all binomial.
The probability of exactly k successes is the binomial probability mass function:
P(X = k) = C(n, k) · pk · (1 − p)n − k
The term C(n, k), read "n choose k," is the binomial coefficient — the number of distinct ways to arrange k successes among n trials. The pk factor is the probability that those k particular trials all succeed, and (1 − p)n − k is the probability the remaining trials all fail. Cumulative probabilities add up the pmf: P(X ≤ k) sums the terms for 0 through k, and P(X ≥ k) equals 1 − P(X ≤ k − 1).
The binomial distribution has a mean of np and a variance of np(1 − p). So for 10 coin flips with p = 0.5, you expect np = 5 heads on average, with a variance of 2.5 (standard deviation about 1.58). The mean is the balance point of the bar chart; the variance describes how wide the spread of likely outcomes is.
For n = 10, p = 0.5, k = 4: C(10, 4) = 210, so P(X = 4) = 210 × 0.54 × 0.56 = 210 / 1024 ≈ 0.2051. The chance of 4 or fewer heads is about 0.3770, and the chance of 4 or more is about 0.8281. As n grows large, the binomial distribution starts to look like a normal distribution — a result known as the normal approximation.
It is the probability distribution of the number of successes in a fixed number of independent trials, where each trial has the same probability of success. The classic example is counting heads in a set number of coin flips. It is defined by two parameters: n (the number of trials) and p (the probability of success on each trial).
Four conditions, often called Bernoulli trials: a fixed number of trials, exactly two possible outcomes per trial (success or failure), a constant probability of success across all trials, and independence between trials. If any condition fails — for example, drawing cards without replacement changes the probability each draw — a different distribution applies.
Use P(X = k) = C(n, k) × p^k × (1 − p)^(n − k). Compute the binomial coefficient C(n, k) for the number of arrangements, multiply by p raised to the number of successes, and by (1 − p) raised to the number of failures. This calculator evaluates that formula exactly and also sums it to give cumulative probabilities.
P(X = k) is the probability of exactly k successes — a single term of the formula. P(X <= k) is the cumulative probability of k or fewer successes, found by adding the terms for 0 through k. P(X >= k) is the probability of k or more, equal to 1 minus P(X <= k − 1). This tool reports all three at once.
The mean (expected number of successes) is np, and the variance is np(1 − p). For example, 100 trials at p = 0.3 have a mean of 30 successes and a variance of 21, giving a standard deviation of about 4.58. These let you summarize the distribution without listing every outcome.